#482689

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,27,28,17)(14,18)(15,21,30,39)(20,35)(22,36)(23,40)(24,33,25,32)(26,34)(29,38,31,37);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,18,30)(12,23,26,20)(13,21,28,39)(16,31,34,29)(17,19,27,36)(24,38,25,37)(32,35,33,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,20)(13,25)(14,26)(15,29)(17,33)(18,34)(19,35)(21,38)(22,23)(24,28)(27,32)(30,31)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(9,13)(10,17)(11,21)(12,24)(14,28)(15,19)(16,32)(18,27)(20,37)(22,39)(23,38)(25,26)(29,40)(30,36)(31,35)(33,34)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=di,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389885);